What is int (sin x)/(cos^3 x) dx ?

1 Answer
Jan 10, 2016

=1/2 sec^2(x)+C or =1/2 tan^2(x) +C

Explanation:

int sin(x)/cos^3(x) dx

Let us use the substitution method.

We can see that the derivative of cos(x) is -sin(x)

Let cos(x)= u
Differentiating with respect to x we get
-sin(x)dx = du
sin(x)dx = -du

Our integral becomes

int (-du)/u^3
=-int u^-3 du

=-(u^(-3+1)/(-3+1))+C

=-u^-2/-2+C
=1/(2u^2)+C

Substituting back

=1/(2cos^2(x))+C

We can also write this as
=1/2 sec^2(x)+C

Note this can also be written as 1/2 tan^2(x) +C
If you are wondering why? It is nothing but replacing sec^2(x) as 1+tan^2(x). Let me show how.

=1/2(1+tan^2)x+C
=1/2+1/2(tan^2(x)+C
=1/2tan^2(x)+C since C is a constant and 1/2+C would be a constant too. We can write it as C itself.

For integration involving trigonometric functions there are usually more than one way of giving an answer so just watch out.